Page 246 - Modern Control Systems

220 Chapter 3 State Variable Models
Obtain the transfer function G(s) = Y(s)/U(s) and x = ax + bu
determine the response of the system to a unit step y — cx + du
input.
where a, b, c, and d are scalars such that the transfer
E3.22 Consider the system in state variable form
function is the same as obtained in (a).
x = Ax + Bit E3.23 Consider a system modeled via the third-order dif-
y = Cx + Du ferential equation
with 'x\t) + 3x(0 + 3x(0 + x{t)
"3 2' " 1 '
A = ,B = , C = [1 0], and D = [0]. = u(t) + 2ii(t) + 4ii(t) + (t).
b 4J L-iJ
Develop a state variable representation and obtain a
(a) Compute the transfer function G(s) = Y(s)/U(s). block diagram of the system assuming the output is
(b) Determine the poles and zeros of the system, (c) If x(t) and the input is u(t).
possible, represent the system as a first-order system

PROBLEMS

P3.1 An RLC circuit is shown in Figure P3.1. (a) Identify P33 An RLC network is shown in Figure P3.3. Define
a suitable set of state variables, (b) Obtain the set of the state variables as x^ = i L and x 2 = v c. Obtain the
first-order differential equations in terms of the state state differential equation.
variables, (c) Write the state differential equation.
Partial answer:
0 \/L
-A/W- A =
L •1/C -l/(RC)
R
v(t) +
Voltage ^ y
source
L ~j[
FIGURE P3.1 RLC circuit.
©
P3.2 A balanced bridge network is shown in Figure P3.2.
(a) Show that the A and B matrices for this circuit are
- 2 / ( ( ^ + R 2)C) 0 FIGURE P3.3 RLC circuit.
0 -2R lR 2/((R l + R 2)L)_
P3.4 The transfer function of a system is
1/C 1/C
B = 1/(/^ + R 2) 2
IR 2/L -R2JL] Y(s) s + 2s + 10
T(s) 2
(b) Sketch the block diagram. The state variables are R(s) ~ s* + 4s + 6s + 10'
(x h x 2) = (v c, i L).
Sketch the block diagram and obtain a state variable
model.
P3.5 A closed-loop control system is shown in Figure
P3.5. (a) Determine the closed-loop transfer function
T(s) = Y(s)IR(s). (b) Sketch a block diagram model
for the system and determine a state variable model.
P3.6 Determine the state variable matrix equation for the
circuit shown in Figure P3.6. Let Xj = V\, x 2 = V2, and
JC3 = i.
P3.7 An automatic depth-control system for a robot sub-
FIGURE P3.2 Balanced bridge network. marine is shown in Figure P3.7.The depth is measured